Past exam of the MPhil in micro- and nanotechnology enterprise course of the University of Cambridge / 2017 / Paper 1 / Section A(iii)

The solid electrolyte conducts oxide ions. Equilibrating oxygen at the two interfaces couples one ${\mathrm{O}}_2$ molecule to four electrons. Taking the positive measured electromotive force as the air-reference potential relative to the Ni/NiO side, the Nernst equation isAssume air at one bar with $p_{\mathrm{air}}/p^{\circ }=0.21$, pure solid phases of unit thermodynamic activity, and $p^{\circ }=1\mathrm{bar}$. For the reaction normalized to one mole of oxygen,The negative sign is physically consistent with a strongly reducing Ni/NiO equilibrium relative to air. Using the supplied $R$ and $F$ givesThe tabulated free energies are per mole of ${\mathrm{O}}_2$, not per mole of NiO. Values normalized to one mole of NiO would be half as large. A reliable oxygen concentration cell measurement requires reversible interfaces, calibrated uniform temperature, a known reference oxygen pressure and predominantly ionic rather than electronic conduction through yttria-stabilized zirconia. Air is not pure oxygen: dropping the $RT\ln 0.21$ term changes the inferred entropy.

Approximate the reaction enthalpy and entropy as constant over the measured interval, so ${\Delta }_r{G}^{\circ }={\Delta }_r{H}^{\circ }-T{\Delta }_r{S}^{\circ }$. A linear least squares fit of all three free energies against temperature givesand henceThe negative entropy change is reasonable because oxygen gas is incorporated into solids. The fit residuals are $+0.129,-0.257,+0.129\mathrm{kJ}\mathrm{mo}{\mathrm{l}}_{{\mathrm{O}}_2}^{-1}$, so the data support this approximate straight-line description. Extrapolating its intercept to zero temperature is a way of estimating a constant local enthalpy, not evidence that the heat capacities vanish or that the same phases persist to zero temperature. More temperatures and repeated heating/cooling measurements would test curvature and equilibration.

A thermoelectric generator maintains hot and cold junctions of two materials with different Seebeck coefficients. Mobile charge carriers redistribute in the temperature gradient, producing an open-circuit voltage of magnitude $|V|=|S_p-S_n|\Delta T$ for constant coefficients. A load permits electric current; pairs of p-type and n-type legs are commonly connected electrically in series and thermally in parallel. Maintaining the output requires continuous heat transfer from the hot reservoir and rejection of heat to the cold reservoir. The thermoelectric figure of merit $ZT=S^2\sigma T/\kappa$ expresses the favorable combination of large Seebeck coefficient, large electrical conductivity and small thermal conductivity. Contact losses, Joule heating and parasitic heat leaks reduce practical efficiency.

A pyroelectric generator instead exploits a change of spontaneous electric polarization with temperature. If the relevant component of the pyroelectric coefficient is $p=d{P}_s/dT$, an electrode of area $A$ receives a short-circuit electric current approximately $I=pAdT/dt$. A heating stroke and a cooling stroke produce opposite charge changes. Rectification or a synchronized electrical cycle can transfer energy to storage. A spatial temperature difference drives the thermoelectric device; a temporal temperature change drives the pyroelectric device. Uniform, constant temperature gives no continuing current from pyroelectricity even if the material is hot. Reported nanowire generators deliberately use changing temperature, rather than infer continuous generation from a static hot specimen.^[ref]

Both approaches can be compact, silent and free of macroscopic moving parts. The thermoelectric generator gives continuous output under a sustained gradient, but needs two effective thermal contacts and sufficient heat flow; very small $\Delta T$ gives a small voltage and thermal contacts can consume much of the available gradient. Thin films exhibiting pyroelectricity can respond to intermittent heating without a permanently maintained hot/cold separation across their thickness, but require repeated thermal cycling, low electrical leakage and suitable charge extraction. Their thermal time constant limits useful cycling frequency, and sensible heating/cooling can dominate the heat budget. A steady hot pipe with an available cold sink favors a thermoelectric generator; periodic radiative heating can favor a pyroelectric generator. Neither converts equilibrium thermal motion at a single fixed temperature into sustained useful work.